Magnitude Difference Calculator
Convert the difference between two stellar magnitudes into a brightness ratio. Essential for comparing how much brighter or dimmer one celestial object is relative to another.
About this calculator
The astronomical magnitude scale is logarithmic and runs counter-intuitively: lower magnitudes mean brighter objects. The brightness ratio between two objects is given by Pogson's Law: ratio = 10^(0.4 × (m₂ − m₁)), where m₁ and m₂ are the magnitudes of the two objects. A difference of 1 magnitude corresponds to a brightness ratio of about 2.512 (the fifth root of 100). A difference of 5 magnitudes equals exactly a factor of 100 in brightness. This logarithmic relationship was formalized by Norman Pogson in 1856 to match the ancient scale set by Hipparchus, where first-magnitude stars were roughly 100 times brighter than sixth-magnitude stars.
How to use
Compare Sirius (magnitude −1.46) to Polaris (magnitude +1.98). Enter −1.46 as First Magnitude and +1.98 as Second Magnitude. The formula computes: ratio = 10^(0.4 × (1.98 − (−1.46))) = 10^(0.4 × 3.44) = 10^1.376 ≈ 23.8. Polaris is therefore about 23.8 times dimmer than Sirius. Swap the inputs to get the reciprocal ratio (~0.042), meaning Sirius is ~23.8 times brighter than Polaris.
Frequently asked questions
How do I calculate the brightness ratio between two stars using magnitudes?
Use Pogson's Law: brightness ratio = 10^(0.4 × Δm), where Δm is the magnitude difference (m₂ − m₁). A positive result greater than 1 means the second object is fainter than the first. For example, a magnitude difference of 5 gives 10^(0.4×5) = 10^2 = 100, confirming the classical rule that a 5-magnitude gap equals a 100-fold brightness difference. This calculator performs that exponentiation automatically.
Why does a smaller magnitude number mean a brighter star?
The magnitude scale originates with the ancient Greek astronomer Hipparchus, who ranked visible stars from 1 (brightest) to 6 (faintest). When photometry made the scale quantitative in the 19th century, astronomers preserved that convention rather than invert it. As a result, very bright objects like Venus or the full Moon have negative magnitudes. The scale is purely a historical artifact, but it is deeply embedded in observational astronomy and is used universally today.
What is the faintest magnitude detectable by the human eye versus a telescope?
Under ideal dark-sky conditions, the unaided human eye can detect stars down to about magnitude +6.5. A modest 10 cm aperture telescope pushes that limit to roughly magnitude +13, while the Hubble Space Telescope has imaged objects fainter than magnitude +31. Each doubling of telescope aperture gains approximately 1.5 magnitudes, and moving from an urban site to a dark site can recover 2–3 magnitudes of limiting depth. Light pollution is therefore one of the greatest practical obstacles in visual and amateur CCD astronomy.